Let The MGF Of { X$}$ Be Given As { M_X(\lambda) = E^{5 \lambda + 7 \lambda^2}$}$. Define A New Random Variable { Z = 3X + C$}$.Which Of The Following Are True For The MGF Of { Z$}$,

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Let the MGF of {X$}$ be given as {M_X(\lambda) = e^{5 \lambda + 7 \lambda^2}$}$.

Definition of Moment Generating Function (MGF)

The Moment Generating Function (MGF) of a random variable {X$}$ is defined as {M_X(\lambda) = E(e^{\lambda X})$}$, where {\lambda$}$ is a real number. The MGF is a powerful tool used to study the properties of a random variable.

Given MGF of {X$}$

The given MGF of {X$}$ is {M_X(\lambda) = e^{5 \lambda + 7 \lambda^2}$}$. This is a quadratic function of {\lambda$}$.

Definition of New Random Variable {Z$}$

A new random variable {Z$}$ is defined as {Z = 3X + c$}$, where {c$}$ is a constant.

MGF of {Z$}$

To find the MGF of {Z$}$, we need to find {M_Z(\lambda) = E(e^{\lambda Z})$}$. Substituting {Z = 3X + c$}$ into the expression, we get:

{M_Z(\lambda) = E(e^{\lambda (3X + c)})$}$

Using the properties of exponents, we can rewrite the expression as:

{M_Z(\lambda) = E(e^{3 \lambda X + \lambda c})$}$

Since {c$}$ is a constant, we can take it out of the expectation:

{M_Z(\lambda) = e^{\lambda c} E(e^{3 \lambda X})$}$

Now, we can use the definition of MGF to rewrite the expression:

{M_Z(\lambda) = e^{\lambda c} M_X(3 \lambda)$}$

Properties of MGF

The MGF has several important properties that we can use to study the properties of a random variable. Some of the key properties are:

  • Linearity: The MGF of a linear combination of random variables is the product of their MGFs.
  • Homogeneity: The MGF of a random variable scaled by a constant is the MGF of the original random variable evaluated at the scaled value.
  • Additivity: The MGF of the sum of two independent random variables is the product of their MGFs.

True Statements about MGF of {Z$}$

Based on the properties of MGF, we can make the following statements about the MGF of {Z$}$:

  • Statement 1: The MGF of {Z$}$ is {M_Z(\lambda) = e^{\lambda c} M_X(3 \lambda)$}$.
  • Statement 2: The MGF of {Z$}$ is a quadratic function of {\lambda$}$.
  • Statement 3: The MGF of {Z$}$ is equal to {M_X(\lambda)$}$ evaluated at ${3 \lambda\$}.

False Statements about MGF of {Z$}$

Based on the properties of MGF, we can also make the following statements about the MGF of {Z$}$:

  • Statement 4: The MGF of {Z$}$ is equal to {M_X(\lambda)$}$ evaluated at {\lambda + c$}$.
  • Statement 5: The MGF of {Z$}$ is a linear function of {\lambda$}$.

Conclusion

In conclusion, the MGF of {Z$}$ is {M_Z(\lambda) = e^{\lambda c} M_X(3 \lambda)$}$. The MGF of {Z$}$ is a quadratic function of {\lambda$}$ and is equal to {M_X(\lambda)$}$ evaluated at ${3 \lambda\$}. The statements about the MGF of {Z$}$ are based on the properties of MGF and are used to study the properties of a random variable.

References

  • Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous Univariate Distributions. Wiley.
  • Papoulis, A. (1984). Probability, Random Variables, and Stochastic Processes. McGraw-Hill.
  • Ross, S. M. (2010). Introduction to Probability Models. Academic Press.

Keywords

  • Moment Generating Function (MGF)
  • Random Variable
  • Quadratic Function
  • Linear Combination
  • Homogeneity
  • Additivity
  • Linearity

Category

  • Mathematics
  • Probability Theory
  • Statistics
    Q&A: Moment Generating Function (MGF) and Random Variables

Q1: What is the Moment Generating Function (MGF) of a random variable?

A1: The Moment Generating Function (MGF) of a random variable {X$}$ is defined as {M_X(\lambda) = E(e^{\lambda X})$}$, where {\lambda$}$ is a real number.

Q2: What is the significance of the MGF in probability theory?

A2: The MGF is a powerful tool used to study the properties of a random variable. It is used to find the moments of a random variable, which are used to describe its distribution.

Q3: How is the MGF of a linear combination of random variables related to the MGFs of the individual random variables?

A3: The MGF of a linear combination of random variables is the product of their MGFs.

Q4: What is the relationship between the MGF of a random variable and its distribution?

A4: The MGF of a random variable is a function that encodes information about its distribution. By analyzing the MGF, we can infer properties of the distribution, such as its moments and shape.

Q5: How is the MGF used in statistics?

A5: The MGF is used in statistics to estimate the parameters of a distribution and to test hypotheses about the distribution.

Q6: What is the difference between the MGF and the characteristic function?

A6: The characteristic function is a related concept to the MGF, but it is defined as {\phi_X(\lambda) = E(e^{i \lambda X})$}$, where {i$}$ is the imaginary unit.

Q7: Can the MGF be used to find the probability density function (PDF) of a random variable?

A7: Yes, the MGF can be used to find the PDF of a random variable. By taking the inverse Fourier transform of the MGF, we can obtain the PDF.

Q8: How is the MGF used in engineering and economics?

A8: The MGF is used in engineering and economics to model and analyze complex systems, such as communication networks and financial markets.

Q9: What are some common applications of the MGF?

A9: Some common applications of the MGF include:

  • Risk analysis: The MGF is used to estimate the probability of extreme events, such as financial crashes or natural disasters.
  • Communication networks: The MGF is used to model and analyze the performance of communication networks.
  • Financial markets: The MGF is used to model and analyze the behavior of financial markets.

Q10: What are some common misconceptions about the MGF?

A10: Some common misconceptions about the MGF include:

  • The MGF is only used for continuous random variables: The MGF can be used for both continuous and discrete random variables.
  • The MGF is only used for symmetric distributions: The MGF can be used for both symmetric and asymmetric distributions.

Conclusion

In conclusion, the Moment Generating Function (MGF) is a powerful tool used to study the properties of a random variable. It is used to find the moments of a random variable, which are used to describe its distribution. The MGF is used in a wide range of applications, including risk analysis, communication networks, and financial markets.

References

  • Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous Univariate Distributions. Wiley.
  • Papoulis, A. (1984). Probability, Random Variables, and Stochastic Processes. McGraw-Hill.
  • Ross, S. M. (2010). Introduction to Probability Models. Academic Press.

Keywords

  • Moment Generating Function (MGF)
  • Random Variable
  • Probability Theory
  • Statistics
  • Risk Analysis
  • Communication Networks
  • Financial Markets

Category

  • Mathematics
  • Probability Theory
  • Statistics